# Formula for a summation

#### mech-eng

Problem Statement
I wonder if there is a formula for this summation: $\sum_{n=1}^5= \frac 1n$
Relevant Equations
I know some formulas for summations but I don't know any formula for this case

$\sum_{k=1}^n=\frac{n({n+1})}2$
I look though some algebra and calculus books but I didn't see any formula for this some, and I am stuck here. I can just represent it in a notation but I cannot think a formulation to obtain the result.

$\sum_{k=1}^{n=5}=\frac {n!}{n!k}$

Thank you.

Last edited:
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#### Ray Vickson

Science Advisor
Homework Helper
Dearly Missed
I look though some algebra and calculus books but I didn't see any formula for this some, and I am stuck here. I can just represent it in a notation but I cannot think a formulation to obtain the result.

$\sum_{k=1}^{n=5}=\frac{n!}{n!k}$

Thank you.
Since you have a simple sum of 5 numbers, what is preventing you from just doing the addition? Admittedly, you need to find a common denominator, but that should not be too hard.

In general, there is no known "closed-form" formula for the so-called harmonic number, defined as
$$H(n) = \sum_{k=1}^n \frac 1 k$$
However, there are simple approximate formulas whose performance becomes better as $n$ becomes larger.

#### mech-eng

Since you have a simple sum of 5 numbers, what is preventing you from just doing the addition?
It was just an example. Sum might be 20 or 30 numbers. Yes with just 5 numbers it is very easy and the common denominator could be 5!. Is that called an $\textrm {harmonic sum}$?

Meanwhile would you also explain why my fraction line does not appear in my post the first post? What is wrong with my latex code?

Thanks

#### WWGD

Science Advisor
Gold Member
Maybe a way of double-checking if the formula is right is using the fact that it is known that the sum will never be an integer.

#### Mark44

Mentor
I wonder if there is a formula for this summation: $\sum_{n=1}^5= \frac 1n$
I know some formulas for summations but I don't know any formula for this case

$\sum_{k=1}^n=\frac{n({n+1})}2$
I look though some algebra and calculus books but I didn't see any formula for this some, and I am stuck here. I can just represent it in a notation but I cannot think a formulation to obtain the result.

$\sum_{k=1}^{n=5}=\frac {n!}{n!k}$
None of your equations makes any sense, since you aren't including that thing being summed.
It's as if you asked someone to evaluate this integral: $\int_1^5$.

Is the first summation supposed to be $\sum_{n=1}^5 \frac 1n$? If so, I don't know of any formula, but it's pretty easy to add the five fractions.

For your second equation, it looks like what you meant is $\sum_{k=1}^n k =\frac{n({n+1})}2$, the sum of the first k positive integers.

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